ارزيابي ارتعاش تيرها با استفاده از چند‌جمله‌‌يي‌هاي متعامد مُفسر

Vibration Assessment of the Beams via Characteristic Orthogonal Polynomials

در اين مطالعه، معادلات ديفرانسيل حاكم بر حركت تير با نظريه‌هاي اولر- برنولي، تيموشنكو و مرتبه- بالاتر تحت اثر جرم متحرك براساس اصل هميلتون به‌دست آمده است. به همين منظور، با استفاده از توابع شكل چندجمله‌يي‌هاي متعامد مفسر و توابع مثلثاتي سازگار با شرايط مرزي، نسبت به تبديل معادلات حاكم به معادلات ديفرانسيل معمولي در حوزه‌ي زمان اقدام شده است. براي بررسي درستي نتايج، بسامد‌هاي ارتعاش آزاد تير با نظريه‌هاي مختلف براي شرايط مرزي ساده و گيردار حاصل از روش پيشنهادي با نتايج ارايه‌شده توسط ساير پژوهشگران مورد مقايسه قرار گرفته است. همچنين، پاسخ ديناميكي تيرها با نظريه‌هاي مختلف تحت اثر بارگذاري جرم متحرك به ازا مقادير متفاوت لاغري تير پايه به‌دست آمده و با نتايج ساير مطالعات مقايسه شده است.

A major issue facing structural engineers is assessing the effects of dynamic loads on structural systems, including beams. The importance of this matter arises from moving vehicles, such as cars and trains, on bridge structures, which are usually simulated by beam structures. Hence, in several studies, in order to explore the dynamic response of beam structures under excitation of dynamic loads, various analytical and numerical methods have been utilized. In this study, to examine easier and faster procedures for finding the dynamic response of Euler-Bernoulli, Timoshenko and Higher-Order beams, a simple semi-analytical method, based on characteristic orthogonal polynomials and trigonometric functions compatible with boundary conditions, is presented. To this end, discrete equations of motion are derived for the three mentioned theories, due to a moving mass, according to the Hamilton principle. Then, the governing equations are transformed into ordinary differential equations in the time domain, and by applying an approximate method, the displacement field of the beam is achieved. In order to consider the efficiency, convergence rate and accuracy of this method, two numerical examples are provided to compare the results of this paper with those presented by other researchers. In this regard, in the former, the free vibration frequencies of the beam with various theories for different boundary conditions were obtained, and it is shown that the results of all three theories give a good convergence rate and high accuracy. Furthermore, in the latter example, the dynamic response of the beams subjected to a moving mass for different values of base beam slenderness was achieved and compared with other studies. Analysis of the maximum dynamic response of the beam and the time history diagrams Showed that the obtained results are in close agreement with those obtained from the numerical method; despite using the lower number of shape functions.